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10.2.2.4. Schmidt-Cassegrain telescope
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10.2.2.4.2. SCT focusing errors
► 10.2.2.4.1. Schmidt-Cassegrain off-axis aberrationsSince Schmidt corrector doesn't induce appreciable abaxial aberrations, in Schmidt-Cassegrain telescopes they are induced by the two mirrors alone. Off-axis aberrations in an SCT - particularly coma and field curvature - can be significant. Although both can be corrected with certain design parameters satisfied, this is not the case with most commercial types, which opt for acceptable optical quality, minimized production cost and broadest commercial appeal. For instance, coma correction would require either aspherizing one of the mirrors, or compromising the ultra-compact design. The former has been tested in commercial units in the recent years (Meade) with not much of a success, mainly due to the overall production quality still being mediocre. In general, SCT astigmatism is low, but coma is typically strong. Follow relations for determining specific values of these aberrations. ● lower-order coma P-V wavefront error for object at infinity is given by: Wc = 2csαd3/3 (115) with the system coma aberration coefficient cs being the sum of the individual aberration coefficients for the two mirrors, cs=c1+c2, and α the field angle. Not surprisingly, there is direct similarity with all-reflecting two-mirror systems, the only difference being in the position of aperture stop. The individual coma coefficients for SCT primary and secondary, respectively - with the secondary's coefficient adjusted for the difference in diameter, so that the two can be directly added - are given by:
with σ1 being the primary mirror stop separation in units of the primary radius of curvature (with the stop at the corrector, σ1=-sc/R1, sc being the corrector-to-primary separation, positive in sign), m the secondary magnification, and
is the secondary mirror stop separation (i.e. exit pupil, or image of the aperture stop formed by the primary to secondary separation), in units of the secondary's radius of curvature (pupil-to-secondary separation is numerically negative for the aperture stop located inside primary's focus); as before, ρ=R2/R1. In a typical ƒ/2/10 commercial SCT (i.e. with secondary magnification m~5), k~0.25, σ1~0.4, ρ~0.31 and σ2~-7.6. This gives needed secondary conic for zero coma as:
Alternative expression for needed secondary conic for zero coma using aggregate parameters can be written as K2=[(BA+1)A-(2-2σ1)C)]/(1-B), where A=(m+1)/(m-1), B=(1-2σ1)k and C=[m/(m-1)]3. Needed σ1 value for zero coma in the arrangement with two spherical mirrors is σ1=[2C-(kA+1)A]/2(C-kA2). Standard commercial SCT has both mirrors spherical, thus K1=K2=0. Obviously, coma in such arrangement is not corrected; for the linear field, it is approximately at the level of an ƒ/6 paraboloid, with the coma increasing to the diffraction-limited level (0.80 Strehl) at about 2.5mm off-axis. SCT system satisfying Eq. 115.2 is so called aplanatic SCT (Fig. 110b). It recently became commercially available advertised as the "improved Ritchey-Chrétien". While admittedly with less astigmatism than comparable RC, it is - needless to say - still "only" an aplanatic SCT. Another option for correcting the coma is to keep the mirrors spherical, but move the corrector (i.e. stop) farther away from the primary. Taking the common ~ƒ/2/10 system with k~0.25, ρ~0.31 and m~5 gives secondary stop separation σ2~0.4-[1.6/(1-2σ1)]. Substituting these values in the relation for secondary's aberration coefficient (Eq. 116) gives c2=0.288σ1-0.528 (omitting common denominator R1) which, after setting c2=-c1 thus 0.288σ1-0.528=-(1-σ1), gives needed corrector separation for corrected coma as σ1=0.663, or nearly 2/3 of the mirror radius of curvature. ● lower-order astigmatism P-V wavefront error for object at infinity is: Wa = asα2d2 (117) with the system astigmatism aberration coefficient as being the sum of the two individual mirror coefficients for the primary and secondary, as=a1+a2, with:
While generally low in the typical compact commercial SCT, astigmatism still requires attention. Relatively small design changes can result in significantly increased astigmatism level. For instance, typical ƒ/2/10 SCT with spherical primary, σ1=0.4 and k=0.25 would, in an aplanatic arrangement, require secondary conic K2=-0.77 for cancelled coma; its astigmatic P-V error would be W=-0.000155(αd)2. Change of the secondary magnification to m=4, corrector separation σ to 0.375 and relative marginal ray height on the secondary k to 0.3 (for identical back focus distance), would require zero-coma secondary conic K2=-1.176 and would have the P-V wavefront error of astigmatism W=-0.000374(αd)2 - greater by a factor of 2.4 (despite that, best image surface curvature would slightly improve, due to the lower Petzval curvature). ● field curvature; of interest are Petzval field curvature, which is not affected by the corrector and, thus, remains as for any two-mirror system, 1/RP=(2/R2)-(2/R1), and best image surface curvature which is, in the presence of astigmatism, given by 1/Rm=(1/RP)+4as or:
In regard to misalignment sensitivity, the SCT, expectedly, has elements of both, two-mirror telescope and a Schmidt camera. In a typical SCT arrangement secondary mirror is mounted on the corrector. If they are centered with respect to each other, their decenter (i.e. lateral shift) with respect to the optical axis produces error of similar magnitude but of opposite sign on their two respective surfaces (Eq. 109 and Eq. 91.2 for the corrector and secondary, respectively), making Schmidt-Cassegrain with spherical mirrors relatively insensitive to smaller decenter errors. The effect is only a slight shift of the best focus from the field center; some negligible residual astigmatism is also present. In an aplanatic SCT, however, (K~-0.8 for conventional f/2/10 system) secondary mirror decenter induces more of coma, making it about three times more sensitive to decenter error. Also, there is no possibility to compensate for decenter error by adjusting secondary tilt; these systems have to be well centered. Tilt of the corrector/secondary, due to negligible error at the corrector, results in center field coma induced by the secondary, as given with Eq. 91.1. Again, in an SCT with spherical mirrors, due to the presence of system coma, at some point off-axis, in the direction of tilt, the two coma contributions will nearly cancel out, and the original field quality can be restored by making plane of the secondary parallel to that of the primary by tilting the former. Same applies to an aplanatic SCT. Since coma due to tilt or lateral surface shift originates from surface deviation, it doesn't change with the field angle. It is added equally to the entire field, which leaves the field center with the amount of coma given by corresponding equations, while subtracting as much coma from one side of the field, and adding it to the other, in the orientation coinciding with that of the corrector's shift. As a result, best focus point shifts off center, in the direction where the misalignment coma is subtracted. The amount of shift depends on the level of SCT coma: in a typical 8" SCT with spherical mirrors, coma induced by as little as 0.2° tilt of the corrector/secondary approximately equals system's coma at 0.25° off-axis. For larger SCTs with spherical mirrors, coma increases in proportion to the aperture diameter, and the shift angle per mm decreases accordingly. In an aplanatic SCT, there is no system coma, and both decentered and tilted corrector/secondary result in coma evenly distributed across the field. Decentered secondary also induces image tilt, which can be significant in an aplanatic SCT (less so in the standard model, due to its field already compromised by strong coma), and can't be corrected by tilt-collimation. Despace sensitivity of the SCT corrector is practically zero; since in practice it supports the secondary mirror, its axial shift would cause error appropriate to mirror decenter, as detailed in SCT focusing errors. Of course, any combination in misalignment of the two mirrors and corrector is possible. EXAMPLE: Taking the typical commercial 8" ƒ/2/10 SCT configuration, with D=8, d=4, R1=-32, k=0.25, m=5 and σ1=0.4, both mirrors spherical, Eq. 112 gives the spherical aberration coefficients s1=-0.000007629 and s2=0.000002197 for the primary and secondary, respectively. This determines needed relative corrector power as P=(s1+s2)/s1=0.712. With the relative exit pupil separation for the secondary mirror σ2=-7.6, the mirror coma coefficients are c1=0.00058594 and c2=-0.0004031 for the primary and secondary, respectively, resulting in the system coma aberration coefficient cs=c1+c2=0.00018282 and the P-V wavefront error of coma Wc=0.0078α, with the RMS wavefront error given by ωc=Wc/√32=0.00138α. With the RMS wavefront error for 0.80 Strehl ωc=λ/√180= 0.0000016136 for λ=550nm=0.00002165", the field angle α at which coma reaches this level is, α=0.0000016136/0.00138=0.00117 radians, or 4 arc minutes. This is somewhat smaller angular field than that of an ƒ/5 parabola. However, the linear field is doubled at ƒ/10, thus having the field appearance of an ~ƒ/6 paraboloid. Secondary conic needed to cancel lower-order coma is K2=-0.77. Ray tracing gives slightly stronger coma, the result of added low-level higher-order coma of the same sign (consequently, needed conic to cancel coma is slightly higher). The astigmatism aberration coefficients are a1=-0.01125 and a2=0.01849 for the primary and secondary, respectively, giving the system aberration coefficient as=a1+a2=0.00724, for the P-V wavefront error Wa=0.11584α2. For α=0.00117 radians, it gives Wa=0.000000158, or 1/137 wave for λ=550nm=0.00002165". Entirely negligible at the field height where the coma reaches 0.80 Strehl level and, for all practical purposes, for the rest of useable field as well. Petzval field curvature of the system is 1/RP=(2/R2)-(2/R1)=-0.14, or -7.16", and the best image surface curvature is 1/Rm=1/RP+4as=-0.11 or -9". Finally, the spherochromatism P-V wavefront error is smaller vs. unit power corrector by a factor of 0.712. From Eq. 106, taking n=1.518, s=0.25 (neutral zone at 0.707 radius), the index differential ι=0.0036 and 0.0044 for the F- and C-line respectively, gives 1/8.1 wave (F) and 1/8.9 waves (C) of spherochromatism at best focus. In terms of the blur size, it is ~1.8 and ~1.6 times their respective Airy disc diameter for the F- and C-line, respectively. The usual neutral zone placement at 0.866 radius results in twice smaller blur, and double the wavefront error. A few words about Schmidt-Gregorian arrangement. In the arrangement with two spherical mirrors it can have coma cancelled with the corrector placed in the proximity of primary's focus (with the secondary protruding in front of it). That would make it even somewhat more compact that coma-corrected SCT, but the drawbacks of significantly stronger astigmatism and nearly doubled spherochromatism (due to contributions from the two mirror being of the same sign) make such a system less attractive.
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